The reduced partition-function conjecture for compressed partitions

Let $\tilde{\boldsymbol{ối}}$ be a partition equal to its compressed partition, and let $\tilde{\boldsymbol{ối}}$ index the multispecies totally asymmetric zero range process (mTAZRP) state space $\tilde{\boldsymbol{ối}}$. Its partition function is the modified Macdonald polynomial

$$\widetilde{H}_{\tilde{\boldsymbol{\bối}}}(X;1,t).$$

Reduced partition-function conjecture. The partition function of this mTAZRP chain is reduced.

This conjecture asserts the expected cancellation of the common factor in the stationary weights for compressed partitions. It is supported by verification for many partitions and system sizes, but the paper does not give a general proof; the analogous statement is noted to fail for the multispecies ASEP.